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Best, Jon On Sat, Feb 10, 2024, at 6:28 AM, David Roberts wrote: > Hi all, > > So here's the thing: my setup is fibrations (non-cloven in general)=20 > over a given model of algebraic set theory, with anafunctors over the=20 > base built using functors preserving cartesian morphisms in the usual=20 > way. So I'm avoiding choice both internal to the model of AST, as well=20 > as externally, so that eso + ff faithful functors aren't always=20 > equivalences in the 2-category of fibrations over the given base. > > I presume B=C3=A9nabou thought at some point about distributors between= =20 > fibrations, but I'd have to dig through those old scanned documents to=20 > see if there is a paper record. My approach would presumably be a=20 > special case of that, as he so forcefully pointed out to me a bit over=20 > a decade ago on this list. If anyone recalls anything along these=20 > lines, I'd appreciate a pointer. > > All the best, > David > > > > David Roberts > Webpage: https://protect-au.mimecast.com/s/BO5DCQnM1WfWPql9SPfE1j?domain= =3Dncatlab.org=20 > > Blog: https://protect-au.mimecast.com/s/0ymoCRONg6sLWl0QuPWulZ?domain=3Dt= hehighergeometer.wordpress.com=20 > > > > On Sat, 10 Feb 2024 at 06:25, Dusko Pavlovic wrote: >> we are all departing, of course. there is no power law there. >>=20 >> the way i understand plack's statement about the paradigm changes when p= eople depart is that some people leave behind houses for other people to li= ve in, whereas others leave behind fortresses with nothing to defend. there= is a paper by thomas kuhn about "The Function of Dogma in Scientific Resea= rch" which i think speaks to us more clearly than his famous "Scientific Re= volutions". >>=20 >> -- dusko >>=20 >> On Thu, Feb 8, 2024 at 3:48=E2=80=AFPM Michael Barr, Prof. wrote: >>> We're departing, we're departing. Just give us a few more years. >>>=20 >>> Michael >>> *From:* Dusko Pavlovic >>> *Sent:* Thursday, February 8, 2024 7:02 PM >>> *To:* David Roberts >>> *Cc:* categories@mq.edu.au >>> *Subject:* Re: Fibrewise opposite fibration=20 >>> =20 >>> FWIW, the history of problems around choosing representatives of equiva= lence classes may very well be a completely self-inflicted artifact of acad= emic politics. if it were up to young cantor, sets could be taken with well= -orders and any equivalence class would have a minimal representative. in t= he world where math is done in collaboration with computers, everything *is= * well-ordered, and cantor was right. if a fibration E is computable, the e= quivalence classes of spans that define E^op have minimal representatives a= nd are effectively defined. in lawvere's words: if we don't transition to c= ardinalities but stay in a boolean topos with cantor-bernstein, all surject= ions split effectively. >>>=20 >>> but dedekind convinced cantor that he should yield to the great profess= ors and not assume that the reals can be well-ordered. then dedekind used c= antor's basic construction in Zahlenbericht (with a reference to cantor in = the manuscript, no reference in the published version), whereas cantor spen= t 10 years trying to *prove* that all sets can be well-ordered. enter zerme= lo to edit cantor's collected works, to criticize cantor for well-ordering,= and to packages this problematic idea behind the second-order quantifier i= n his own contribution: the axiom of choice. so that we could happily choos= e the representatives of equivalence classes ever after from behind the sec= ond-order quantifier =3D& >>>=20 >>> but it will be like max planck said: the new paradigm will win when the= generations suppressing it depart. in the world where math is computed, ze= rmelo was wrong, cantor was right, and equivalence classes have minimal rep= resentatives. OR if we develop category theory categorically, the dual hom-= sets will be effectively definable... >>>=20 >>> :) >>> -- dusko >>>=20 >>> On Sat, Jan 27, 2024 at 7:21=E2=80=AFPM David Roberts wrote: >>>> Hi all >>>>=20 >>>> what with all the discussion of B=C3=A9nabou and fibrations, I have a >>>> question about what happens at a foundational level when one takes the >>>> opposite of a fibration E --> B fibrewise. Call this (E/B)^op --> B >>>>=20 >>>> For reference, one can see section 5 of Streicher's >>>> https://protect-au.mimecast.com/s/Xv6nCVARmOHAyKkXHJDjBn?domain=3Darxi= v.org for the construction. The point is >>>> that the morphisms are defined to be equivalence classes of certain >>>> data. However, in a setting where one cannot necessarily form >>>> equivalence classes, it's less clear how to proceed. The point is that >>>> I don't want to be assuming any particular foundations here, just >>>> working at the level of a first-order theory (in the way that ETCS is >>>> a first-order theory of sets, say) >>>>=20 >>>> The only thing I can think of is that the construction actually >>>> describes a category weakly enriched in 0-truncated groupoids (or >>>> whatever you want to call the first-order description of such a >>>> thing). You still get a functor down to the base 1-category, and >>>> perhaps one has to now think about what it means for such a thing to >>>> be a fibration, without passing to the plan 1-category quotient. >>>>=20 >>>> That is probably fine for my purposes, but then you have to worry that >>>> taking the fibrewise opposite again should return the original >>>> fibration, at least up to equivalence. The original construction with >>>> the equivalence classes gives back the original thing up to >>>> *isomorphism*: ((E/B)^op/B)^op \simeq E, over B. So now one has to >>>> think about what the fiberwise opposite construction looks like for >>>> these slightly generalised fibrations (enriched with 0-truncated >>>> groupoids), and one would hope that this gives back the original thing >>>> after two applications (again, up to the appropriate notion of >>>> equivalence). >>>>=20 >>>> Note that the construction in the literature (eg Streicher's notes, or >>>> Jacob's book) has the fibres (E/B)^op_b of the fibrewise opposite be >>>> *isomorphic* to the opposite of the original fibres E_b. In this >>>> fancier setting, one might also only get equivalence, but I haven't >>>> checked that. >>>>=20 >>>> Has anyone thought about something like this before? Or any pointers >>>> to anything related? >>>>=20 >>>> Best wishes, >>>>=20 >>>> David Roberts >>>> Webpage: https://protect-au.mimecast.com/s/BO5DCQnM1WfWPql9SPfE1j?doma= in=3Dncatlab.org >>>> Blog: https://protect-au.mimecast.com/s/0ymoCRONg6sLWl0QuPWulZ?domain= =3Dthehighergeometer.wordpress.com >>>> =20 >>>> =20 >>>> You're receiving this message because you're a member of the Categorie= s mailing list group from Macquarie University. To take part in this conver= sation, reply all to this message.=20 >>>> =20 >>>> View group files | Leave group | = Learn more about Microsoft 365 Groups =20 >>>> =20 >>> =20 >>> =20 >>> You're receiving this message because you're a member of the Categories= mailing list group from Macquarie University. To take part in this convers= ation, reply all to this message.=20 >>> =20 >>> View group files | Leave group | = Learn more about Microsoft 365 Groups =20 >>> =20 >=20 >=20 > You're receiving this message because you're a member of the Categories= =20 > mailing list group from Macquarie University. To take part in this=20 > conversation, reply all to this message.=20 >=20 > View group files=20 > =20 > | Leave group=20 > =20 > | Learn more about Microsoft 365 Groups --MCBoundary=_12402101943090121 Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset=utf-8 Dear David,

In case it helps, there is a nicely typeset paper by Bénabou and Str= eicher about distributors between fibrations here: https://www2.mathematik.tu-darmstadt.de/~streicher/FIBR/DibFi.pdf= .

Best,
Jon


On Sat, Feb 10, 2024, at 6:28 AM, David Roberts wrote:
> Hi all,
>
> So here's the thing: my setup is fibrations (non-cloven in general) > over a given model of algebraic set theory, with anafunctors over the =
> base built using functors preserving cartesian morphisms in the usual =
> way. So I'm avoiding choice both internal to the model of AST, as well=
> as externally, so that eso + ff faithful functors aren't always
> equivalences in the 2-category of fibrations over the given base.
>
> I presume Bénabou thought at some point about distributors betw= een
> fibrations, but I'd have to dig through those old scanned documents to=
> see if there is a paper record. My approach would presumably be a
> special case of that, as he so forcefully pointed out to me a bit over=
> a decade ago on this list. If anyone recalls anything along these
> lines, I'd appreciate a pointer.
>
> All the best,
> David
>
>
>
> David Roberts
> Webpage: https://ncatlab.org/nlab/show/David+Roberts
> <https://protect-au.mimecast.com/s/pAyTCK1DOrCnW5pot= MTgxQ?domain=3Dncatlab.org>
> Blog: https://thehighergeometer.wor= dpress.com
> <https://protect-au.mimecast.com= /s/spbWCL7Eg9f19n8WsqojnV?domain=3Dthehighergeometer.wordpress.com><= BR> >
>
> On Sat, 10 Feb 2024 at 06:25, Dusko Pavlovic <duskgoo@gmail.com>= wrote:
>> we are all departing, of course. there is no power law there.
>>
>> the way i understand plack's statement about the paradigm changes = when people depart is that some people leave behind houses for other people= to live in, whereas others leave behind fortresses with nothing to defend.= there is a paper by thomas kuhn about "The Function of Dogma in Scien= tific Research" which i think speaks to us more clearly than his famou= s "Scientific Revolutions".
>>
>> -- dusko
>>
>> On Thu, Feb 8, 2024 at 3:48=E2=80=AFPM Michael Barr, Prof. <bar= r.michael@mcgill.ca> wrote:
>>> We're departing, we're departing. Just give us a few more yea= rs.
>>>
>>> Michael
>>> *From:* Dusko Pavlovic <duskgoo@gmail.com>
>>> *Sent:* Thursday, February 8, 2024 7:02 PM
>>> *To:* David Roberts <droberts.65537@gmail.com>
>>> *Cc:* categories@mq.edu.au <categories@mq.edu.au>
>>> *Subject:* Re: Fibrewise opposite fibration
>>>
>>> FWIW, the history of problems around choosing representatives = of equivalence classes may very well be a completely self-inflicted artifac= t of academic politics. if it were up to young cantor, sets could be taken = with well-orders and any equivalence class would have a minimal representat= ive. in the world where math is done in collaboration with computers, every= thing *is* well-ordered, and cantor was right. if a fibration E is computab= le, the equivalence classes of spans that define E^op have minimal represen= tatives and are effectively defined. in lawvere's words: if we don't transi= tion to cardinalities but stay in a boolean topos with cantor-bernstein, al= l surjections split effectively.
>>>
>>> but dedekind convinced cantor that he should yield to the grea= t professors and not assume that the reals can be well-ordered. then dedeki= nd used cantor's basic construction in Zahlenbericht (with a reference to c= antor in the manuscript, no reference in the published version), whereas ca= ntor spent 10 years trying to *prove* that all sets can be well-ordered. en= ter zermelo to edit cantor's collected works, to criticize cantor for well-= ordering, and to packages this problematic idea behind the second-order qua= ntifier in his own contribution: the axiom of choice. so that we could happ= ily choose the representatives of equivalence classes ever after from behin= d the second-order quantifier =3D&
>>>
>>> but it will be like max planck said: the new paradigm will win= when the generations suppressing it depart. in the world where math is com= puted, zermelo was wrong, cantor was right, and equivalence classes have mi= nimal representatives. OR if we develop category theory categorically, the = dual hom-sets will be effectively definable...
>>>
>>> :)
>>> -- dusko
>>>
>>> On Sat, Jan 27, 2024 at 7:21=E2=80=AFPM David Roberts <drob= erts.65537@gmail.com> wrote:
>>>> Hi all
>>>>
>>>> what with all the discussion of Bénabou and fibrati= ons, I have a
>>>> question about what happens at a foundational level when o= ne takes the
>>>> opposite of a fibration E --> B fibrewise. Call this (E= /B)^op --> B
>>>>
>>>> For reference, one can see section 5 of Streicher's
>>>> https://arxiv.org/abs/1801.02927 <https://protect-au.mimecast.com/s/tjAKCMwGj8CZ34mrFJj1nS?domain=3D= arxiv.org> for the construction. The point is
>>>> that the morphisms are defined to be equivalence classes o= f certain
>>>> data. However, in a setting where one cannot necessarily f= orm
>>>> equivalence classes, it's less clear how to proceed. The p= oint is that
>>>> I don't want to be assuming any particular foundations her= e, just
>>>> working at the level of a first-order theory (in the way t= hat ETCS is
>>>> a first-order theory of sets, say)
>>>>
>>>> The only thing I can think of is that the construction act= ually
>>>> describes a category weakly enriched in 0-truncated groupo= ids (or
>>>> whatever you want to call the first-order description of s= uch a
>>>> thing). You still get a functor down to the base 1-categor= y, and
>>>> perhaps one has to now think about what it means for such = a thing to
>>>> be a fibration, without passing to the plan 1-category quo= tient.
>>>>
>>>> That is probably fine for my purposes, but then you have t= o worry that
>>>> taking the fibrewise opposite again should return the orig= inal
>>>> fibration, at least up to equivalence. The original constr= uction with
>>>> the equivalence classes gives back the original thing up t= o
>>>> *isomorphism*: ((E/B)^op/B)^op \simeq E, over B. So now on= e has to
>>>> think about what the fiberwise opposite construction looks= like for
>>>> these slightly generalised fibrations (enriched with 0-tru= ncated
>>>> groupoids), and one would hope that this gives back the or= iginal thing
>>>> after two applications (again, up to the appropriate notio= n of
>>>> equivalence).
>>>>
>>>> Note that the construction in the literature (eg Streicher= 's notes, or
>>>> Jacob's book) has the fibres (E/B)^op_b of the fibrewise o= pposite be
>>>> *isomorphic* to the opposite of the original fibres E_b. I= n this
>>>> fancier setting, one might also only get equivalence, but = I haven't
>>>> checked that.
>>>>
>>>> Has anyone thought about something like this before? Or an= y pointers
>>>> to anything related?
>>>>
>>>> Best wishes,
>>>>
>>>> David Roberts
>>>> Webpage: https://ncatlab.org/nlab/show/Davi= d+Roberts <https://protect-au.mimecast.com/s/pAyTCK1D= OrCnW5potMTgxQ?domain=3Dncatlab.org>
>>>> Blog: https://thehigher= geometer.wordpress.com <https://p= rotect-au.mimecast.com/s/spbWCL7Eg9f19n8WsqojnV?domain=3Dthehighergeometer.= wordpress.com>
>>>>
>>>>
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>>>>
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